/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 28 If \(\\{X(t)\\}\) and \(\\{Y(t)\... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 28

If \(\\{X(t)\\}\) and \(\\{Y(t)\\}\) are independent continuous-time Markov chains, both of which are time reversible, show that the process \(\\{X(t), Y(t)\\}\) is also a time reversible Markov chain.

Short Answer

Expert verified
Since both \(X(t))\) and \(Y(t)\) are Markov chains, their joint process \(\\{X(t), Y(t)\\}\) is also a Markov chain by proving the Markov property for any three times \(t_1<t_2<t_3\) and any three pairs \((x_1,y_1)\), \((x_2,y_2)\), \((x_3,y_3)\). As both \(X(t)\) and \(Y(t)\) are time reversible, the joint process satisfies time reversibility property too \(P((X(t+s), Y(t+s)) = (x_1, y_1) | (X(t), Y(t)) = (x_2, y_2)) = P((X(s), Y(s)) = (x_2, y_2) | (X(0), Y(0)) = (x_1, y_1))\). Thus, the joint process \(\\{X(t), Y(t)\\}\) is a time reversible Markov chain.

Step by step solution

01

Prove that the joint process is a Markov chain

To prove that the joint process \(\\{X(t), Y(t)\\}\) is a Markov chain, we need to show that for any three times \(t_1 < t_2 < t_3\) and any three pairs \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\), we have the Markov property: \(P((X(t_3), Y(t_3)) = (x_3, y_3) | (X(t_2), Y(t_2)) = (x_2, y_2), (X(t_1), Y(t_1)) = (x_1, y_1)) = P((X(t_3), Y(t_3)) = (x_3, y_3) | (X(t_2), Y(t_2)) = (x_2, y_2))\) Since we are given that \(X(t)\) and \(Y(t)\) are independent, we have \(P((X(t_3), Y(t_3)) = (x_3, y_3) | (X(t_2), Y(t_2)) = (x_2, y_2), (X(t_1), Y(t_1)) = (x_1, y_1)) = P(X(t_3) = x_3 | X(t_2) = x_2, X(t_1) = x_1)P(Y(t_3) = y_3 | Y(t_2) = y_2, Y(t_1) = y_1)\) Since both \(X(t)\) and \(Y(t)\) are Markov chains, we have \(P(X(t_3) = x_3 | X(t_2) = x_2, X(t_1) = x_1) = P(X(t_3) = x_3 | X(t_2) = x_2)\) and \(P(Y(t_3) = y_3 | Y(t_2) = y_2, Y(t_1) = y_1) = P(Y(t_3) = y_3 | Y(t_2) = y_2)\) Therefore, \(P((X(t_3), Y(t_3)) = (x_3, y_3) | (X(t_2), Y(t_2)) = (x_2, y_2), (X(t_1), Y(t_1)) = (x_1, y_1)) = P((X(t_3), Y(t_3)) = (x_3, y_3) | (X(t_2), Y(t_2)) = (x_2, y_2))\) Hence, the joint process \(\\{X(t), Y(t)\\}\) is a Markov chain.
02

Prove that the joint process is time reversible

Now, we need to show that if \(X(t)\) and \(Y(t)\) are time reversible, then so is their joint process. The joint process will be time reversible if, for any two pairs \((x_1, y_1)\) and \((x_2, y_2)\) and any time \(t\), we have \(P((X(t+s), Y(t+s)) = (x_1, y_1) | (X(t), Y(t)) = (x_2, y_2)) = P((X(s), Y(s)) = (x_2, y_2) | (X(0), Y(0)) = (x_1, y_1))\) We know that \(X(t)\) and \(Y(t)\) are time reversible, so we have \(P(X(t+s) = x_1 | X(t) = x_2) = P(X(s) = x_2 | X(0) = x_1)\) and \(P(Y(t+s) = y_1 | Y(t) = y_2) = P(Y(s) = y_2 | Y(0) = y_1)\) Using the independence of \(X(t)\) and \(Y(t)\), we have \(P((X(t+s), Y(t+s)) = (x_1, y_1) | (X(t), Y(t)) = (x_2, y_2)) = P(X(t+s) = x_1 | X(t) = x_2)P(Y(t+s) = y_1 | Y(t) = y_2) = P(X(s) = x_2 | X(0) = x_1)P(Y(s) = y_2 | Y(0) = y_1) = P((X(s), Y(s)) = (x_2, y_2) | (X(0), Y(0)) = (x_1, y_1))\) Thus, the joint process \(\\{X(t), Y(t)\\}\) is time reversible, proving the given exercise.

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Most popular questions from this chapter

In the \(M / M / s\) queue if you allow the service rate to depend on the number in the system (but in such a way so that it is ergodic), what can you say about the output process? What can you say when the service rate \(\mu\) remains unchanged but \(\lambda>s \mu ?\)

Potential customers arrive at a single-server station in accordance with a Poisson process with rate \(\lambda\). However, if the arrival finds \(n\) customers already in the station, then he will enter the system with probability \(\alpha_{n}\). Assuming an exponential service rate \(\mu\), set this up as a birth and death process and determine the birth and death rates.

Consider two \(M / M / 1\) queues with respective parameters \(\lambda_{i}, \mu_{i}, i=1,2 .\) Suppose they share a common waiting room that can hold at most three customers. That is, whenever an arrival finds her server busy and three customers in the waiting room, she goes away. Find the limiting probability that there will be \(n\) queue 1 customers and \(m\) queue 2 customers in the system. Hint: Use the results of Exercise 28 together with the concept of truncation.

There are two machines, one of which is used as a spare. A working machine will function for an exponential time with rate \(\lambda\) and will then fail. Upon failure, it is immediately replaced by the other machine if that one is in working order, and it goes to the repair facility. The repair facility consists of a single person who takes an exponential time with rate \(\mu\) to repair a failed machine. At the repair facility, the newly failed machine enters service if the repairperson is free. If the repairperson is busy, it waits until the other machine is fixed; at that time, the newly repaired machine is put in service and repair begins on the other one. Starting with both machines in working condition, find (a) the expected value and (b) the variance of the time until both are in the repair facility. (c) In the long run, what proportion of time is there a working machine?

Consider a set of \(n\) machines and a single repair facility to service these machines. Suppose that when machine \(i, i=1, \ldots, n\), fails it requires an exponentially distributed amount of work with rate \(\mu_{i}\) to repair it. The repair facility divides its efforts equally among all failed machines in the sense that whenever there are \(k\) failed machines each one receives work at a rate of \(1 / k\) per unit time. If there are a total of \(r\) working machines, including machine \(i\), then \(i\) fails at an instantaneous rate \(\lambda_{i} / r\) (a) Define an appropriate state space so as to be able to analyze the preceding system as a continuous-time Markov chain. (b) Give the instantaneous transition rates (that is, give the \(q_{i j}\) ). (c) Write the time reversibility equations. (d) Find the limiting probabilities and show that the process is time reversible.
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